Algebroid Mappings and Their Equidistribution Theory

TL;DR

This paper introduces algebroid mappings for complex manifolds and develops an equidistribution theory, extending Nevanlinna theory to analyze multi-valued solutions of complex PDE systems.

Contribution

It proposes a new framework for algebroid mappings and establishes a second main theorem for their equidistribution on Kähler manifolds.

Findings

Established a second main theorem for algebroid mappings.

Extended Nevanlinna theory to complex PDE solutions.

Provided conditions for uniformization of algebroid mappings.

Abstract

In this paper, the concept of algebroid mappings of complex manifolds is introduced based on that a large number of complex systems of PDEs admit multi-valued solutions that can be defined by a system of independent algebraic equations over the field of meromorphic functions. It is well-known that Nevanlinna theory is an important tool in complex ODE theory. To develop a similar tool applied to the study of algebroid solutions of complex systems of PDEs, one explores the equidistribution theory of algebroid mappings. Via uniformizating an algebroid mapping, we obtain a second main theorem of algebroid mappings from a complete K\"ahler manifold into a complex projective manifold provided that some certain conditions are imposed.